Dissociative collinear reactions evaluated by S-matrix propagation along delves' radial coordinate

Volume 77,

number

CHEMICAL PHYSICS LETTERS

I

DlSSOCiATWE

COLLINEAR

EVALUATED

1 January 1981

REACTIONS

BY S-MATRIX

PROPAGATION

ALONG

DELVES’

RADIAL

COORDINATE

J MA&Z Leitrsztrh~ fttr Titcormsclte

Citemre. Trchtttsclte

Dnt! erstrat ~Vrtttcitctt, D SO16 Garclttttg

West CerttzarzJ

and .I ROMELT

Rccewed

1 September

E\act quantum dlssocntw

flon Tllc

probabIhtws

ch~nncls results

19SO

tot 3 colltne~r

and tar enqles

LLN obtamcd

ranymg

b> S-nutrL\

trntorntc trom

model

the re.wt~on

propaystlon

along

re;lctlon

XC cvaluntcd,

threshold

over

Dclvcs’ mdul

rncludmg

the dasoclarlon

non-reactwe threshold

reactwca,rd

to the brcah-up

re-

coordmata

I [ncroduct~on

2 Method

In tius paper. we prtlsenr the first exict quantum waluatron of a cothnea trmtonm dissocmttve reactton

The dtssoczatwe reaction (I a)-( 1c) IS evaluated by stmple routme apphcarron of a novel method that we lmve introduced recently [I ,2] S-matrt\ propagatton (51 along Delves’ radial coordinate The details of the method have already been presented and are dtrectly adapted from refs fl,?J Therefore, here we merely evpkun how we solve the dehcate probiem f3] of the szmultaneotzs occurrence of the dzssoczatrve continuum together wtth dtscrete reactant atzd product channels in processes (I a)-(1 c) Our solution 1s based on the use of Delves’ coordtnates (v, &) ii .6,7] which are tlhzstrated rn fig 1 They describe the rhree processes (la)-( lc) stmukaneously as radial and internal mottons along r and 9, respecttvety. rn the mteractton regton, I e for small w&.zes r, of P, the internal motton ts in a szngle-mtnimum potentzai V(rz, $) Wtth mcreasmg values of r, Il(r,, Ip) ts transformed tnto a double-mtmmum potenttal The two mmtma represent the bound molecular channels (la) and (1 b) They are separated by the plateau. f/(rl, y) = 0, For the dzssociatzon f ic)_ The tnternal motton is always confined to Cl < y <

A + BC@) - A f BC(V’),

(12)

A AB(Y’) + C,

(lb)

-zt-t+B+C

(ICI

tncludmg non-reacrlve (lo), reactw (lb) and dasoctat~ve (I c) channels for enetgtes rangmg from the rcxt Ion threshold (,Y 0) Processes (la\(I c) are most stmply reahzed tn systems wttb small nutnbers of vzbratzonai reactant (v) and product (v’) levels, e g. when A, B and C are iight tnert atoms Ttus sttuatton IS modelied tn sectton 3, and the correspondmg exact quantum probablltttes for reactton and dtssocmtton are presented m sectton 4 Conclustons are tn sectzon 5; but first, zn sectton 2, we present our method [ 12 1 and compare it wth the prevtous exact approaches [3,43 _

9 ma\’ grna, = artan[t?zg(r?zA 172

0 009-26

14/81/0000-0000/S

02.50

+trzg +wzC)/mAmc]

Q North-Holland

F’ublishrng

II2 Company

Volume

7’7, number

i

CHEMICAL

f9S.l

1 JanuarY

PHYSICS LETTERS

1 I31erps &, of Internal motions versus Delves’ ndmi coordmste r (for the modei of sectton 3f_ The arrows A and B nrarb the lourron of the saddle pomt and (rot@& j rhe onset of the dlssacntlon pl;ltesu, respectwely 4s r mcr~ses, the gende (g) and ungerade (u) bound ieseisdegenerare to the reactant (r) and product (pj molcculru Ievels The dwioczt.tr\e JeveJ scheme approaches the square-weli tcsuits l-g

1-g 1 ~q~ll~nto~lrs W, up) = E, and proftlcs t’(t = rl, 9) oi energy surface tor a dtssoclattve colhnmr reactton plotted III Dehes’ wordmates r = (6 -t I~~)“?, 9 = ;ucOS (V/Y) Y = (U~~,BC//?~BC)~‘~ f~,~c,)’ = FBC (schemntlc)

tltc potentwl

Therefore IE is represented by a drscrete basrs set 9, (r,. p) ‘tv~th discrete energies C,, Cr,) wfuch depend paramerrlcslly on r1 (via t/(r,,y}, see fig 7-) Asymptotically. for large values ofr,, the Cp, wth ~5, < 0 and &, > 0 describe channels (la}, (lb) and (Ic), respectively Ultunately, for r, - m the posttlve &” become arbltrarI& dense However, only rat!ler low-u channels fu 5 6 m the present case, cf sect&on 3) are coupled durmg the propagatmn along r Therefore, converged numencal results are obtamed already at fimte values of r, rather close to the inreraction region Thus the dlssoclatlve contmuum s replaced a~tom~tlcaI~y by a rattler smafl discrete level set descrzbrng internal m&tons Irr fact, the numerical treatment of the bound (Ia), (1 b) and dlssoczative channels (1 c) IS sunply identical, The preslous (< 1978) exact approaches to colhon-

mduced dlssocmtlon have been revlewed by DIestIer [3] The techruques used dtd not allow the ~multaneous evaluation of dlssoclatlve (ic) and reactive (1 b) channels The first -and so far tfle only - exact quantum evaluat!on of a dssoclat!ve reaction has been presented by Kulander t4j in lus time-dependent approach, a wavepacket is propagated through the uztenctlon regoon and IS analyzed m the asymptotrc configurations. This cechmque IS very useful if the wavepacket remains rather compact throughout the colhsion, e g , the meth-

od IS taylored to the Important case of fast co&ions between weakly ekctted reactants. As an example, Kulander mvestlgated the system

+ti+H+H,

C2C)

with weakly excrred reactants (v < 4) for h.@t collision energes E = 3-12 eV. The wavepacket then is scattered back almost entn-eiy Into the non-reactive region (However, the results [4] stlfi lack rhe convergence test that dtfferent wavepackets prepared tn overlappmg energy ranges yield consistent results ) An eqtuvajent applicatton to reactions (3a) and (2b) at lower colhnon energtes ISi did not yxeid the standard state- and energy-resolved results, see refs. [P-I 11. tn contrast, our method has been tested [2] by reproducmg perfectiy the results of refs. [9-I I ] An equivalent method has been developed simultaneously but lndependen~ly by Kuppemxann et ai fIZ!J In addition, we have solved the problem of bght-atom exchange reactlons [ 133 employmg the present method Cl ,2].

173

Volume 77. number 1

CHEMICAL

PHYSICS

LETTERS

1

3. The model

(E3

For the first apphcatlon of our method to dlssoclatrve reactlons, we select for su-nphclty a homonuclear model system [ 141 x -i- X,(v)

-7 x f Xz(v’),

(3a)

--f X,(v’)

(3b)

+ x.

+x+x+x,

(3cI

with one vibrational reactant and product level, v = 0 and v’ = 0, respectrvely Thus the attractive mternuclear mteractron IS very weak as for inert atoms In order to mmlc processes (3a)-(3cj, we set J?Z~ = 02~ and employ the Porter-Karplus surface [15] for processes (2a)-(2c) scaled by a factor f = 1000/6. The results are equivalent to those obtained on the orrgmal surface [ 151 wrth masses m X = “zH/f Thus our SySte2Tl models dlssoclatlve reactlons (la)-(lc) of Inert atoms, however, to factitate the comparison with the estabhshed results for the hydrogen reaction (2a)-(2c) (see refs [2,4,9-131). the results are presented below aslf they were obtained for a (hypothetical) mass-scaled lsotoplc variant (3a)-(3c) on the orlgrnal potential energy surface [ 15 ] Accordmg to a classical theorem [16], the results for the dissocratlve reactions (2a)(2~) and (3a)-(3c) should be identical Some dlfferences are expected, however, due to the elact quantum description

4. Results and discussion The resultmg probablhtles for reaction pPR(3b) and dlssoclatlon PDR(3c) for energ es rangmg From the reaction threshold E > - 1 83 eV to the break-up regron E < 5 eV are presented m fig. 3. The non-reactive probablhtres are given by P RR = 1 _ PPR _ PDR, smce there IS only one reactant and one product vibrational level for the sunple model of sectron 3. The reactive probability mcreases rather sharply to its maxlmum value and then rt decreases smoothly towards neghgrble values close to the dlssoclatlve threshold. In the posltlve energy range mvestrgated here, the dissoclatlve probablhty mcreases steadily and becomes dommant over the reactive channel. These findmgs agree quahtatlvely with the famlhar results of the hydrogen reac174

1

5 I

I

DI / eV 7 I I

January 1981

9

1 d lssoclatlon

I I

f

\ reactlon d; I

I-

I I

\i

E /eV Fig 3 Exact quantum probablhtres for reactlon - and dlssoclatmn o versus energy E (or E - D, D = -4 7466 eV [15 1) for the model system of sectlon 3 The dasocntlve threshold IS mdlcated by the dashed Ime

tlon (2b) for E < 0 [g-12] and dlssoclatlon (2~) for E > 0 [4] (v = 0 case). Quantltatlve equivalence of colhs~ons (2a)-(2c) and (3a)-(3c) IS predicted on classlcal grounds [16], but the quantum nature of the systems introduces important differences in their reactlvltles and resonance patterns. In the energy range considered, the dissociative probablhty remains rather small, in agreement with prevrous classical results for the colhnear colhslon Induced dlssoclatlon Ar + Ar, + 3Ar [17J. 5. Conclusions The present paper demonstrates that dissoclatlve reactions (la)-( 1c) can be routmely evaluated by our novel techmque [ 1,2], S-matrrx propagation along Delves’ radial coordinate For the first time, rt comprehends the restricted ranges of apphcations of all previous approaches, includmg reactrve and dlssoclatrve channels and covering the energres below and above the &ssoclatrve threshold_ The success of the method and Its numerical sun-

Volume

77, number

1

CHEMICAL

PHYSICS

pllcity are based on Delves’ coordinates [6,7] they descrrbe aZCprocesses - the non-reactive (1 a), reactrve (1 b) and dissociat rve (1 c) ones - as ooze pseudo-melastrc colhmon In alternative, more techmcal terms, several adequate hamdtoman representatrons, one for each asymptotrc channel, are umfied mto ONE for aii con~guratlons [ l] The present progress made wrth our method it ,2] may be vrewed III synopsis wrth prevrous sohnrons of dehcate problems which also employ Delves’ coordtnates [6,7] the statrstrcal theory of colhsron-Induced drssocratton [18,19], the evaluatron of the hydrogen reactions (2a) and (2b) up to the Y = 6 threshold f12], and of lightatom exchange reactions [ 131. Strlughtforward extensrons of the present work wII rnciude total and drfferentrai drssocratwe reaction probabthtres, heteronuclear systems, and rnelastrc channels

Acknowledgement We shoufd Ike to thank Professor D J_ Drestler for suggestmg the model of section 3; Dr KC. Kulander for strmulatmg correspondence, Professor A Kuppermann for sending us hrs preprmt [12] prror to pubhcatron; and Professors S D. Peyerunhoff and R _I Buenker for theu support. The financial support through a NATO project and by the Fonds der chemlschen Industrre ts also gratefully ac~o~viedged The computations have been carried out at the RHRZ at the Untversrty of Bonn

LETTERS

1 January

1981

References [I]

G Hauke, J hfanz and J Romelt, J. Chem Phys , to be pubhshed [Z] J. Romelt, Chem Phys Letters 74 (1980) 763 131 D J. DiestIer, m Atom-molecule colhslon theory, ed R 3 Bernstem (Plenum Press. New York, 1979) pp 65.5-667 [4] KC Kulander, J Chem Phys 69 (1978) 5064 [51 J Manz, Mel Phys 28 (1974) 399 [6j LXI Delves,Nucl Phys 9 (1959) 391 [7 ] L M Delves, Nucl Phys 20 (1960) 275. [S] E A McCullough Jr and R E Wyatt, J Chem Ph)s 54 (197 1) 3592 [9) DJ. DiestIer, J Chem Phys 54 (1971) 4547 phys Letters 13 (1972) 172 1101 B R Johnson,Chem flil 3 C Lght and R 3 WaIker. J Chem Phys 65 (1976) 4272 1121A Euppermann, J A. Knye and J P Dwyer, Chem Phys Letters 74 (1980) 757 [I31 J hlanz and J Rome&. Chem Phys Letters 76 (1980) 337 (1980) iI41 D J DIestIer, prwate commumutlon I151 R N Porter and BI Karplus, J Chem. Phys 40 11964) 1105. 1161 P J Kuntz, E hf Nemeth, 3 C Polanyl, S D Rosner and C E Young, J Chem Phys 44 (1966) 1168 Roberts, J Chem 1171 hf DeI!eDonne, R. Howard and RE Phys 64 (1976) 3387 II81 R D Levme and R B Bernstem, Chem Phys Letters II (1971) 552. [I91 C Reblck and R_D Levrne, J Chem Ph1.s 58 (1973) 3942

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